Thursday, November 08, 2012

How not to Teach Math—or Economics

A recent conversation with my younger son, frustrated over his undergraduate math course, reminded me of my long standing objection to how math, and for that matter economics, are often taught. Theorems are proved with a rigor that is more than the students really need—especially in economics, where rigorous proofs can be applied to the real world only by combining them with non-rigorous models. The rigor is not only more than the student needs, it is more than any save the ablest students can understand. It is one thing to follow a proof step by step. It is a different and much more difficult thing to hold the proof in your head and understand why it is right.

My usual example of the problem is the failure to teach students of calculus why the fundamental theorem, that integrating and taking a derivative are inverse operations, is true. It is possible to give a non-rigorous but intuitively persuasive proof of the theorem in about five minutes, one that any student who understands what the two operations are can follow and has a good chance of remembering. If any reader is sufficiently skeptical to warrant the effort, I can do it here.

In my experience, very few of the students who take calculus, even at a good school, are ever shown the proof; I would be surprised if more than one in fifty, a year after taking the course, could reproduce the more rigorous proof that they were, presumably, taught. To check the former impression, I asked my wife for her experience. Her response was that she was taught calculus twice, the first time at a good suburban school (but by an incompetent teacher), the second time at a top liberal arts college. To the best of her memory, she was never shown the proof.

I take as a further piece of evidence the math problems that my son was asking my help with—having assured me that the rules did allow him to discuss homework problems with people. One of them was a problem so trivial that a bright ten year old could probably have solved it without having taken the course, provided only that the problem itself was clearly explained to him. I take the fact that such a problem was assigned as evidence that the instructor believes a substantial fraction of the students do not understand what they are being asked to do, quite aside from knowing how to do it.

It is common, at good schools, to complain against "cookbook mathematics," the sort of course that consists of memorizing the sequence of steps to solve a problem without ever understanding why it works. But it is, I think, an equally serious mistake to present a branch of mathematics in the form in which professional mathematicians structure it after all of the original work in that particular field is done. Not only is it a form in which almost no student not qualified to become a professional mathematician can understand it, it is a form that gives a highly misleading picture of how mathematics, or other forms of theory, are actually done.

I am not a mathematician but I am an economist and know by direct observation how the original parts of my work were done. The process did not start with a step by step proof but with an intuition of how some set of ideas fit together, what characteristics the solution to a problem ought to have. Only after I had groped my way to what was (hopefully) the right answer did I, or someone else, go back and make the argument rigorous.

I am currently working on the third edition of my first book, published about forty years ago. One of the things I am doing is filling in the blanks, working out in more depth and more detail ideas whose essence I understood then and still believe, in most cases, were correct.

Alfred Marshall, arguably the figure most responsible for the creation of neo-classical economics, commented in a letter on the relation between mathematical and verbal arguments in his field. He explained that he worked his arguments out mathematically to make sure they were right. Having done so, he translated them into English. If he could not translate them into English, he burned the mathematics.

There is much to be said for that policy. Mathematics is a more precise language than English for the sort of work Marshall was doing. But it is also a language farther from the intuition of almost all of us. If you have the math and cannot translate it, it is quite likely, although not certain, that the reason is you do not understand it. I sometimes referee journal articles. Occasionally I get one where, if you translate the math into words, it makes no sense—is arguably insane. The author or authors presumably had doctorates in the field. But they were manipulating symbols, not ideas.

My daughter is at the same college as her brother. When she transferred there after two years elsewhere, she was seriously considering majoring in economics. After taking an economics course, she decided on her other alternative major. The reason was not that she does not like economics, or cannot do it—she audited several of my courses while a home schooled student of high school age, and one of my articles contains an idea that I credit in a footnote to her, since it was hers.

The reason was that the course was mostly about the math not the economics. I discussed her experience with a professor at that university of whom I have a high opinion—someone on the short list of people who, when they disagree with me, cause me to seriously consider that I may be making a mistake. He also had a daughter taking economics at the same school. He agreed with my daughter's judgement—that the courses were teaching mathematical rigor instead of economic intuition.

I have let this essay wander from my son's experience to my daughter's by way of mine. But I think the fundamental thread is the same in all. What matters is not remembering but understanding. If you have learned a proof but cannot explain why the result is true, you have been wasting your time.

34 Comments:

I had always looked at calculus as something completely incomprehensible until I stumbled acrossMartin L. Bittinger (Author of Calculus and it's Applications). It was not as rigorous as I need later when I was at Polytechic Insitute of NY University but it was a really great introduction. The thing it left out that i wish had been there was the Taylor series.

I have a bright 10 year old available. Would it be possible to see the math problem you mentioned to test your idea?

Regarding calculus, my understanding of it improved substantially after I studied courses in other subjects(computer science, physics, statistics and economics) where various features of calculus were used. Therefore my general theory is: we shouldn't teach math as such, but we should integrate it in other subjects as needed.

I appreciate David's attitude, but disagree. I believe it true that Alfred Marshall's "English" and the thinking behind it are to a great extent informed by mathematical understanding. To me, mathematics comprises things like formal logic and sentence diagramming, and it is the lack of mathematical rigor, indicating a lack of understanding, that is so evident in the bad speech and writing that we are exposed to daily.

I always wonder what's going on in Obama's mind or Bill Clinton's mind when he says things like:

"The reason is, is that....""Between Hillary and I...." "Everybody ... they....""The media is ....""...grow a business...."

Obama and Bill Clinton, though accomplished men, must have been mentally absent during 7th-grade sentence diagramming. Were they also behind the barn when Darwin handed out logical thinking? Maybe in a Muslim academy in Indonesia? Or a Creationism school in Arkansas?

Other grammatical abortions that we are exposed to daily include:

"more complete," "more unique," "more saturated," "more perfect." What is going on in the mind of a person who says things like, "In order to establish a more perfect union...."? Would a person say such things if he had once been asked to locate "more saturated" or "more perfect" on a graph? Can one snowflake be "more unique" than another? Can a set be "more complete" than another?

The problem is is that our country is more than saturated by dolts who, like Obama, haven't a clue as to either math or English, much less how to run a country. But, as they say, "I could care less...."!

Well, I am an odd duck because I don't feel like I understand anything until I've seen a mathematically rigorous presentation of it, definition-theorem-proof style. I guess that's why I decided to study mathematics.

Most fields of science, not just mathematics, suffer from this dichotomy between rigorous modern theory and the actual historical development of the ideas. It's just more pronounced in math because the modern theory part is on a higher level than physics or chemistry or medicine. How many physicists have read translations of Galileo or Newton? How many chemists ever read Boyle's or Lavoisier's lab notes? On calculus the book "The history of calculus and its conceptual deveopment" is a standard historical text you may like.

The solution for non-math students is simple: Give incomplete proofs. An incomplete proof leaves out some details but is usually good enough. Of course you must be clear about the difference.

The solution for students in math (or chemistry, physics, etc) is not so simple. Do math students need to read the parts of "The Elements" where odd*odd=odd and even*even=even are painfully proved? Do chemistry students need to read about phlogiston, or physics students about the detailed setup of the Michelson-Morley experiment? Most engineering students could not tell you how to build a steamboat or how Herbert Dow refined bromine. (So if civilization collapses good luck restarting it with a bunch of semi-conductor and aeronautic engineers!)

You said you are familiar with the original parts of your work, but are you really? Have you read Aquinas and Hume, Hutchinson and Smith, Ricardo and Marx, Menger and Malthus, Mill and Marshall, Hayek, Mises, Keynes, Sraffa, and Galbraith? How much would it profit students to learn about the original creative ideas in their field, and the vastly more numerous examples of ideas that were eventually abandoned or refuted?

These questions may seem inane, but as you imply they are directly relevant to modern instruction. Why? Because people are still coming up with creative ideas today! So then the difficult questions are these: What mix of creative original thought and rigorous presentation is the proper one? In how many research journals can the readers summarize the latest articles in english, or be able to replicate the results that were published without cheating by contacting the authors for more information? In how many math or computer journals can the result by result through a computer and checked for accuracy, or can be easily explained in english to an undergraduate student in 15 minutes?

And how many go wrong? In how many journals are the articles filled with useless technical jargon, or in how many are they filled with flowery prose trying to awkwardly explain an equation?

I've been taught undergraduate mathematics in France, which has a tradition of rigor in the field. For instance, our first the course about matrices came after the course about endomorphisms, vector spaces, fields, rings and initially groups. Every theorem was rigorously demonstrated, and exercised consisted in demonstrating properties of the objects studied.

I for one greatly enjoyed this approach to mathematics. I wouldn't say it is incompatible with getting an intuitive grasp of what actually happens. In fact it's probably impossible to remember the proofs without it.

For instance, to prove that the primitive is the inverse operation of the derivative. One can start with Rieman's definition of the derivative, draw little rectangles under the curve, and show that derivating the cumulative integral is equivalent in the limit to looking at the height of the curve on that point. Having said that, the proof immediately follows by equating the difference between two Riemann sums and Newton's ratio.

I think the real problem is that students are taught "calculus" instead of "analysis". I was baffled when I learned that students are taught to do formal derivation ( sin -> cos, x^n -> nx^(n-1), etc ) without understanding why. I understand that high-school students are given exercises where they have to derive functions and yet have no idea why this works. This strikes me as the root of the problem.

" Have you read Aquinas and Hume, Hutchinson and Smith, Ricardo and Marx, Menger and Malthus, Mill and Marshall, Hayek, Mises, Keynes, Sraffa, and Galbraith?

Some Aquinas, some Hume, most of Smith, Ricardo's Principles, some Marx, some Menger, a little Malthus, a little Mill, all of Marshall's principles, a little Hayek, almosst no Mises, some Keynes, no Sraffa, some Galbraith.

Of those, I learned some economics from Smith, Ricardo and Marshall.

But I think you are missing my point. I'm not arguing for learning the history of how the field developed. I'm arguing for learning the subject in a way that gives you a more accurate picture of how the field develops, which is an entirely different thing. Ricardo alone, or Marshall alone, would do that. Or a good course focusing on intuition.

"How much would it profit students to learn about the original creative ideas in their field, and the vastly more numerous examples of ideas that were eventually abandoned or refuted?"

Arthur sketches my proof in one sentence, although it would take the drawing to make it clear to someone not already familiar with the subject.

I agree that "it's probably impossible to remember the proofs" without the intuition. But I doubt that more than a few percent of undergraduates do remember the proofs a year after the course. For the small minority who can manage both intuitions and rigor I have no objection to doing so.

I got it over the phone. You have an infinite series of terms, positive and decreasing. Take the sum of the first n terms, with alternating signs:

A1-A2+A3-A4 ...

You are told that there is a number L which is in between every consecutive pair of sums-- between the sum (with alternating signs) of the first six and the first seven terms, the first 20 and the first 21, etc. for all of them.

Prove that the sum of the first n terms never differs from L by more than the next term--An+1.

(I'm not sure if the problem specified that the terms were of decreasing magnitude or not. If my son is reading this he can correct any misstatement--I'm going by memory).

Depending on the mathematical sophistication of your ten year old, you can translate my description into terms he can understand.

But I think you are missing my point. I'm not arguing for learning the history of how the field developed. I'm arguing for learning the subject in a way that gives you a more accurate picture of how the field develops, which is an entirely different thing. Ricardo alone, or Marshall alone, would do that. Or a good course focusing on intuition.

I guess I don't understand the distinction. Any field develops by people coming with ideas, most of which don't work, and then over time discovering or proving a few that do work. By focusing only on Marshall, for example, you're excluding discussion of ideas that failed, and perhaps antecedent ideas Marshall himself used to come up with his theory. Isn't that relevant if Marshall is?

Put another way, it seems to me you're just sweeping away the difficulties of improving rigor with intuition and the progression of thought. I don't want to have to teach students about phlogiston, but how else can you give an accurate description of how the ideas developed? At a minimum, you would have to at least discuss the ideas and observations that Marshall drew on. Otherwise it seems like just drawing an arbitrary line at a certain date or author, in which case why not draw it at a modern textbook? That leaves us back at where we started.

I also disagree with the idea of providing a separate course for this. If intuition and the progression of ideas are separated from rigor, one winds up with history courses that students can't connect with the main courses. If this is going to work at all, it has to be in tandom with normal rigorous work, not because of it. For an example of what I mean, see "Classical Dynamics of Particles and Systems" by Jerry Marion. The preface mentions that one reason people like this book is the number of historical footnotes interspersed in the text which help give students an idea of the history usually lacking in other courses.

There is a distinction between teaching the history of the field and teaching the way the ideas in the field were developed, but it's a fine one. The former might be interpreted as "we're going to teach these topics just because somebody came up with them several hundred years ago," whereas the latter uses history to teach students the actual process of advancing knowledge, complete with false starts and dead ends.

I'm not convinced that 49 out of 50 students are inherently incapable of understanding a rigorous mathematical proof, but the approach described above isn't really for the benefit of such students anyway -- it's really for the benefit of those who will specialize in, and advance, the field.

I think a majority of the mathematicians I work with favor that approach, at least at the lip-service level. There are a couple of difficulties implementing it in practice:

(1) some of the proofs and solutions that people have come up with in the past are Just So Cool And Clever that it's hard to resist the temptation to show them to students.

(2) There's a limited amount of time in a course to cover a prescribed list of topics. Teaching those topics complete with the false starts and dead ends takes a lot longer, so it's hard to cover all the things you want to cover.

(3) As a result of (1) and (2), many existing textbooks present mathematical topics through a rear-view mirror, as we see them now rather than the way their originators saw them. I'm always happy to find a textbook that illustrates a new topic by starting with a couple of mistakes before showing a solution that actually works, but that's only one of several factors I weigh in choosing a textbook.

I agree that only a small few who learned how to prove mathematical theorems are able to repeat the exercise many years later, but wonder if that's a sufficient argument for not testing for that proficiency in the first place.

I have an (unrecognized) minor in math, and I don't recall having been taught how to prove the theorems we applied during the four calculus courses I took as an undergraduate. I do remember there was at least one, possibly two, more advanced courses specifically devoted to that.

I took calculus at UC San Diego, back in 1968-1969, and of course we studied the fundamental theorem. But I never really had an intuitive sense for why it was true; it was something to memorize. Then years later, when I was tutoring someone in calculus—I think it was surfaces and volumes of revolution—it hit me that, well, obviously the rate at which the area under a curve increased just had to be proportional to the height of the curve. And so I was enlightened, as they say in Zen.

Much more recently, in thinking about classical economics, I've worked out ways to prove Say's Law (by mathematical induction on the number of commodities) and Ricardo's Law (by applying the theory of convex bodies). In both cases, the mathematics just came to me from struggling to make sense of an example. That moment of seeing that it just has to be a certain way is the really important one, in my modest experience.

I think you are both misunderstanding me. I'm not saying anything at all about teaching the history of the field, although I do find history of economic thought interesting and have taught it. I'm saying that one ought to teach students in a way that gives them a clearer picture of how the people who develop a field actually think--which isn't by proving theorems, as a rule.

If a hen-and-a-half lays an egg-and-a-half in a day-and-a-half, how many eggs do three hens lay in three days?

Eitan, it seems not to matter that the ant walks on the outside of the cube. The answer is obvious from symmetry considerations and in any case can be found experimentally by attaching a rubber band externally to the two opposing vertices and snapping it.

I'm sorry to be obtuse, but I'm still unsure what you're saying. The example you chose, calculus, is hundreds of years old, and Marshall is a century old. So the way the "people who develop the field" think is historical in these cases, because Newton, Leibniz, and Marshall developed those things and are historical figures.

Are you talking about replacing proofs with convincing arguments? I sometimes hear teachers push that line of reasoning, saying proofs are abstruse and students hate them and the results are intuitively clear anyway. In fact that sounds exactly like what you're saying here:

That happens after you first figure out that the theorem is true.

The problem with that is there's a huge gap between proof and convincing argument. Compare Euclid with all his flaws to the exemplar of the convincing argument - Aristotle. Intellectuals believed in a lot of the brilliant reasoning in Aristotle for thousands of years, but sometimes it was proven wrong (as the in the four elements, or heavier objects falling faster than light ones). I know of no theorem in "The Elements" that has ever been shown to be false. That is a strong case for accepting proof over convincing argument, and in fact "The Elements" was the standard textbook in math until modern times.

Eitan,1) Consider any distribution of 5 points on the sphere. Two of these points define a great circle--let that be the equator. Each of these points is in both the northern and southern hemisphere. By the pigeonhole principle, at least two of the remaining three points must be in one of those two hemispheres. That's four.

(If being on the equator does not count as being in either hemisphere, then you can get a counterexample to your theorem by taking a great circle, inscribing a regular pentagon, and calling its vertices your set of 5 points.)

2) The shortest distance the ant can travel is a path along two faces, 1 by .5 and then .5 by 1. You can prove this this is the shortest path by symmetry, or by setting the derivative of sqrt(1+x^2)+sqrt(1+(1-x)^2) to zero. The total distance is sqrt(5).

Good job guys. I think about the cube one by flattening the cube into the 6 squares in the shape of a cross which would fold up to make the cube. Then just think about a straight line (hence, shortest path) across two adjacent squares connecting their opposite corners. By Pythagoras' theorem, the length is sqrt(1^2 + 2^2)=sqrt(5).

I think you are both misunderstanding me. I'm not saying anything at all about teaching the history of the field.... I'm saying that one ought to teach students in a way that gives them a clearer picture of how the people who develop a field actually think--which isn't by proving theorems, as a rule.

That happens after you first figure out that the theorem is true.

I agree entirely, and I thought that's what I was arguing for too: not "teaching the history of the field", but "teaching the reasoning that researchers in the field have historically followed." Which, as you say, usually involves proving things rigorously only after you have good reason to believe that they're true.

I once interviewed for a job at St. John's College in Annapolis, which is all about reading the primary sources -- not just to know the history, but to understand how the founders of the field thought while they were founding the field.

hudebnik: I agree also, and thought that's what I was arguing for as well on the history, which is why David's objection didn't make sense to me. Certainly you don't want to list dates or birthdays or talk about Einstein's marriages or Newton's gruffness; you want the ideas and thoughts relevant to the discovery of the theories. It's obvious to me anyway, but maybe I missed something? Do "history of X" courses really indulge in extensive and superfluous biography?

I don't want to talk about Smith's birthdays, but I also don't want (in this context) to talk about his ideas. I want to teach economics and math in something closer to the form in which people who come up with original ideas in the field do it. That doesn't require saying anything at all about the history of ideas.

I have a friend who has a BS in Math but is now studiing economics. I have seen the problems she was given for her exam preparations by her prof; they were various multi-dimensional optimizations under a linear (budget) constraint. Essentially, students were given a multivariate (saym n dimensiona) utility function and asked to find the maximum on an n-1 dimensional (hyper-) plane.She could do it (having studied math before), but she was profoundly puzzled how this stuff relates to economics.I think that the far more important skill is transforming real-life challenges into such problems. Solving them is a trivial skill that is not even necessary for economists to master to the point of being able to do it fast.While I have only one semester of undergraduate microeconomics behind me (being a telecom engineer and an applied mathematician by call and training), I thoroughly enjoyed analyzing missile defense with those tools:http://www.epointsystem.org/~nagydani/missdef.pdfI actually know (and acknowledge) that the math behind it is not rigorous, but given the assumptions, making it rigorous would be a waste of effort and highly misleading.

I want to teach economics and math in something closer to the form in which people who come up with original ideas in the field do it. That doesn't require saying anything at all about the history of ideas.

So you want teachers to come up with new arguments and intuitions for, say, calculus, which are very like the original work they currently do, and unlike the actual original arguments that Newton employed? That might work for some fields like economics, but I don't see how it would work for math or physics. Newton and Einstein, for example, had some pretty clear and intuitive ideas that underlay their achievements.

Dr David:I've always thought that engineers, physicists etc should be taught Calculus in their own depts. Unfortunately, Calculus is regarded as a hurdle course - if you can't learn it the way it's always been taught, you can't become one of us.

I think we should start by teaching a course called "Calculus Problems" - heavily geometric, using approximate derivatives and integrals until the concept of limit is highly motivated and the student knows how to get answers reliably.

Then teach limits. If proofs would be informative, teach a couple but prefer results. Teach the fundamental theorem (result - not proof) just to amaze.

At this point, if a rigorous course in theorem proofs is needed, teach "Proofs of Calculus" to the few. Let the scientists who want to use Calculus for physics escape the semester of proofs.

I have no objection to using Newton's intuition if that's the best way of teaching an idea. But using an intuition that has been used by Newton is not the same thing as teaching it because it was used by Newton.

And if I was teaching the Pythagorean theorem, I would use the simple "square inside a square proof," not the Euclidean one, since it is a whole lot easier to understand and remember.

I think it's telling that you immediately brought up the Pythagorean theorem as an example. That proposition has literally hundreds of proofs to choose from, and the most intuitive can be picked by your taste. But how many ideas of Newton, Einstein, or the myriad others in science have dozens, or even more than one, way to discover, understand, and explain its development? I wonder if you would still use calculus as an example after reading "The History of the Calculus and Its Conceptual Development"?

Ultimately, what I think you're after is a way to roughly and without rigor explain ideas to students in these fields in such a way that it feels motivated and plausible. The best ways of doing that make the student think they could have found the idea themselves with sufficient effort. The great thing about using history is that one person definitely found a chain of causally-connected ideas leading to a theorem, like Newton did with calculus. The way you're proposing sounds like you'd often wind up with a sequence of ad hoc ideas which wouldn't be any more motivated than the original (correct) rigorous proof.

I'm replying late,sorry about that. While I completely agree that intuition is the important thing if the goal is an understanding of a specific field (calculus in 1 of your examples) I also feel that it's a pretty worthy goal to teach people about rigourous proofs in general. This is really the business of pure mathematics. I certainly can't remember even a quarter of the proofs I've learned but having learned them probably does help me in proving new theorems. I realize that yeah most people are not going to become mathematicians but I'm a little hesitant to dismiss the idea of showing undergrads rigourous proofs at all. Because I think it's as or more valuable to learn about the idea of proof in general as it is to learn calc. Yes I admit that the current way of teaching calc doesn't help them understand proofs that well either but I'd be careful of chucking the idea of learning rigourous proofs aside too quickly.

I should probably mention that I am a mathematics major (finnishing Ms degree in probability theory this year and hopefully finding a sutiable PhD position afterwards).

That said I agree with pretty much everything David Friedman said in his article - I might have something to add to it:

First of all, doctor Friedman (and others interested in the problems of teaching mathematics or more generally - what mathematics really is), if you have not read it already, take a look at this article : www.maa.org/devlin/lockhartslament.pdf

It is a serious critique of the current basic/high school curriculum in the US (which I am not from, but the situation is exactly the same in the czech republic) and shows why what is presented there as mathematics really has nothing to do with it (partly because of being "cookbook" and partly for too much rigor..basically in line with Friedman's objections but giving wonderful comparisons to paiting or music and develops some ideas further).

I think the ideal way to teach mathematics is to simulate the process of an actual mathematical work:

First outline some problem in an informal way and make an educated guess how that can be treated. Then try to work your way towards that - again in an informal fashion - which gives you the idea what means you might need to achieve your goal. Once you have that, you are almost done. You have constructed an informal proof. The only work left is to make a final check - constructing the rigorous (more or less, as some tedious parts can be omitted if you and your students are confident enough about them) proof.

The final step is to make the thing organized. Formulate the ideas into a statement, make definitions of some newly created constructs (the definitions may play both the role of an intuitive object and a way to describe something rather complex with just one or two words). If you are teaching students whose major is not mathematics (or physics to some extent), stop after making the informal proof, because most of them will not do very serious mathematical work themselves anyway and so they do not need to worry that much about the language.

One last thing to do afterwards:

Give a nice, simple,trivial example of how the theorem works and what it implies. People can understand those with ease and most are able to incorporate it in more complex ideas later - but not if they haven't seen it work in a trivial case in the first place. One thing I really like about physics (which I know very little of) is that physicist are often ready to give very simple examples of how very complex things work. And they help tremendeously in making you understand them (and remmember them as well).

The problem is is that our country is more than saturated by dolts who, like Obama, haven't a clue as to either math or English, much less how to run a country. But, as they say, "I could care less...."!